I Am a Strange Loop

by Douglas R. Hofstadter

  Our own brains are no different from careenia, except, of course, that whereas careenia are just my little fantasy, human brains are not. The symbols in our brains truly do do that voodoo that they do so well, and they do it in the electrochemical soup of neural events. The strange thing, though, is that over the eons that it took for our brains to evolve from the earliest proto-brains, meanings just sneaked ever so quietly into the story, almost unobserved. It’s not as if somebody had devised a grand plan, millions of years in advance, that high-level meaningful structures — physical patterns representing abstract categories — would one day come to inhabit big fancy brains; rather, such patterns (the “symbols” of this book) simply came along as an unplanned by-product of the tremendously effective way that having bigger and bigger brains helped beings to survive better and better in a terribly cutthroat world.

  Just as Bertrand Russell was blindsided by the unexpected appearance of high-level Gödelian meanings in the heart of his ultraprotected bastion, Principia Mathematica, so someone who had never conceived of looking at a brain at any level other than that of Hans Moravec’s squirting chemicals would be mightily surprised at the emergence of symbols. Much as Gödel saw the great potential of shifting attention to a wholly different level of PM strings, so I am suggesting (though I’m certainly far from the first) that we have to shift our attention to a far higher level of brain activity in order to find symbols, concepts, meanings, desires, and, ultimately, our selves.

  The funny thing is that we humans all are focused on that level without ever having had any choice in the matter. We automatically see our brains’ activity as entirely symbolic. I find something wonderfully strange and upside-down about this, and I’ll now try to show why through an allegory.

  In which the Alfbert Visits Austranius

  Imagine, if you will, the small, lonely planet of Austranius, whose sole inhabitants are a tribe called the “Klüdgerot”. From time immemorial, the Klüdgerot have lived out their curious lives in a dense jungle of extremely long PM strings, some of which they can safely ingest (strings being their sole source of nutrition) and others of which they must not ingest, lest they be mortally poisoned. Luckily, the resourceful Klüdgerot have found a way to tell apart these opposite sorts of PM strings, for certain strings, when inspected visually, form a message that says, in the lilting Klüdgerotic tongue, “I am edible”, while others form a message in Klüdgerotic that says “I am inedible”. And, quite marvelously, by the Benevolent Grace of Göd, every PM string proclaiming its edibility has turned out to be edible, while every PM string proclaiming its inedibility has turned out to be inedible. Thus have the Klüdgerot thrived for untold öörs on their bountiful planet.

  On a fateful döö in the Austranian möönth of Spöö, a strange-looking orange spacecraft swoops down from the distant planet of Ukia and lands exactly at the North Pöö of Austranius. Out steps a hulking whiteheaded alien that announces itself with the words, “I am the Alfbert. Behold.” No sooner has the alien uttered these few words than it trundles off into the Austranian jungle, where it spends not only the rest of Spöö but also all of Blöö, after which it trundles back, slightly bedraggled but otherwise no worse for the wear, to its spacecraft. Bright and early the next döö, the Alfbert solemnly convenes a meeting of all the Klüdgerot on Austranius. As soon as they all have assembled, the Alfbert begins to speak.

  “Good döö, virtuous Klüdgerot,” intones the Alfbert. “It is my privilege to report to you that I have made an Austranius-shaking scientific discovery.” The Klüdgerot all sit in respectful if skeptical silence. “Each PM string that grows on this planet,” continues the Alfbert, “turns out to be not merely a long and pretty vine but also, astonishingly enough, a message that can be read and understood. Do not doubt me!” On hearing this non-novelty, many Klüdgerot yawn in unison, and a voice shouts out, “Tell us about it, white head!”, at which scattered chuckles erupt. Encouraged, the Alfbert does just so. “I have made the fantastic discovery that every PM string makes a claim, in my beautiful native tongue of Alfbertic, about certain wondrous entities known as the ‘whole numbers’. Many of you are undoubtedly champing at the bit to have me explain to you, in very simple terms that you can understand, what these so-called ‘whole numbers’ are.”

  At the sound of this term, a loud rustling noise is heard among the assembled crowd. Unbeknownst to the Alfbert, the Klüdgerot have for countless generations held the entities called “whole numbers” to be incomprehensibly abstract; indeed, the whole numbers were long ago unanimously declared so loathsome that they were forever banned from the planet, along with all their names. Clearly, the Alfbert’s message is not welcome here. It is of course wrong (that goes without saying), but it is not merely wrong; it is also totally absurd, and it is repugnant, to boot.

  But the whiteheaded Alfbert, blithely unaware of the resentment it has churned up, continues to speak as the mob rustles ever more agitatedly. “Yes, denizens of Austranius, fabulously unlikely though it may sound, in each PM string there resides meaning. All it takes is to know how to look at the string in the proper way. By using a suitable mapping, one can…”

  All at once pandemonium erupts: has the Alfbert not just uttered the despised word “one”, the long-banished name of the most dreaded of all the whole numbers? “Away with the alien! Off with its white head!” screams the infuriated mob, and a moment later, a phalanx of Klüdgerot grabs hold of the declaiming alien. Yet even as it is being dragged away, the pontificating Alfbert patiently insists to the Klüdgerot that it is merely trying to edify them, that it can perceive momentous facts hidden to them by reading the strings in a language of which they are ignorant, and that… But the angered throng drowns out the Alfbert’s grandiose words.

  As the brazen alien is being prepared to meet a dire fate, a commotion suddenly breaks out among the Klüdgerot; they have plumb forgotten the age-old and venerated Klüdgerot tradition of holding a Pre-dishing-out-ofdire-fate Banquet! A team is dispatched to pick the sweetest of all PM strings from the Principial Planetary Park of Wööw, a sacred sanctuary into which no Klüdgerot has ever ventured before; when it returns with a fine harvest of succulent strings from Wööw, each of which clearly reads “I am edible”, it is greeted by a hail of thunderous applause. After the Klüdgerot have expressed their gratitude to Göd, the traditional Pre-dishing-out-ofdire-fate Banquet begins, and at last it begins to dawn on the Alfbert that it will indeed meet a dire fate in short order. As this ominous fact takes hold, it feels its white head start to spin, then to swim, and then…

  Idealistically attempting to save the unsuspecting Klüdgerot, the ever-magnanimous Alfbert cries out, “Listen, I pray, O friends! Your harvest of PM strings is treacherous! A foolish superstition has tricked you into thinking they are nutritious, but the truth is otherwise. When decoded as messages, these strings all make such grievously false statements about whole numbers that no one — I repeat, no one! — could swallow them.” But the words of warning come too late, for the PM strings from Wööw are already being swallowed whole by the stubbornly superstitious Klüdgerot.

  And before long, frightful groans are heard resounding far and near; the sensitive Alfbert shields its gaze from the dreadful event. When at last it dares to look, it beholds a sorry sight; on every side, as far as its sole eye can see, lie lifeless shells of Klüdgerot that but moments ago were carousing their silly heads off. “If only they had listened to me!”, sadly muses the kindly Alfbert, scratching its great white head in puzzlement. On these words, it trundles back to its strange-looking orange spacecraft at the North Pöö, takes one last glance at the bleak Klüdgerot-littered landscape of Austranius, and finally presses the small round “Takeoff” button on the craft’s leatherette dashboard, setting off for destinations unknown.

  At this point, the Alfbert, having earlier swooned in terror as the banqueters began their ritual reveling, regains consciousness. First it hears shouts of excitement echoing all around, and then, when it dar
es to look, it beholds a startling sight; on every side, as far as its sole eye can see, masses of Klüdgerot are staring with unmistakable delight at something moving, somewhere above its white head. It turns to see what this could possibly be, just in time to catch the most fleeting glimpse of a thin shape making a strange, high-pitched rustling sound as it rapidly plummets towards —

  Brief Debriefing

  I offer my apologies to the late Ambrose Bierce for this rather feeble imitation of the plot of his masterful short story “An Occurrence at Owl Creek Bridge”, but my intentions are good. The raison d’être of my rather flippant allegory is to turn the classic tragicomedy starring Alfred North Whitehead and Bertrand Russell (jointly alias the Alfbert) and Kurt Gödel (alias the Klüdgerot) on its head, by positing bizarre creatures who cannot imagine the idea of any number-theoretical meaning in PM strings, but who nonetheless see the strings as meaningful messages — it’s just that they see only high-level Gödelian meanings. This is the diametric opposite of what one would naïvely expect, since PM notation was invented expressly to write down statements about numbers and their properties, certainly not to write down Gödelian statements about themselves!

  A few remarks are in order here to prevent confusions that this allegory might otherwise engender. In the first place, the length of any PM string that speaks of its own properties (Gödel’s string KG being the prototype, of course) is not merely “enormous”, as I wrote at the allegory’s outset; it is inconceivable. I have never tried to calculate how many symbols Gödel’s string would consist of if it were written out in pure PM notation, because I would hardly know how to begin the calculation. I suspect that its symbol-count might well exceed “Graham’s constant”, which is usually cited as “the largest number ever to appear in a mathematical proof”, but even if not, it would certainly give it a run for its money. So the idea of anyone directly reading the strings that grow on Austranius, whether on a low level, as statements about whole numbers, or on a high level, as statements about their own edibility, is utter nonsense. (Of course, so is the idea that strings of mathematical symbols could grow in jungles on a faraway planet, as well as the idea that they could be eaten, but that’s allegoric license.)

  Gödel created his statement KG through a series of 46 escalating stages, in which he shows that in principle, certain notions about numbers could be written down in PM notation. A typical such notion is “the exponent of the kth prime number in the prime factorization of n”. This notion depends on prior notions defined in earlier stages, such as “exponent”, “prime number”, “kth prime number”, “prime factorization” (none of which come as “built-in notions” in PM). Gödel never explicitly writes out PM expressions for such notions, because doing so would require writing down a prohibitively long chain of PM symbols. Instead, each individual notion is given a name, a kind of abbreviation, which could theoretically be expanded out into pure PM notation if need be, and which is then used in further steps. Over and over again, Gödel exploits alreadydefined abbreviations in defining further abbreviations, thus carefully building a tower of increasing complexity and abstractness, working his way up to its apex, which is the notion of prim numbers.

  Soaps in Sanskrit

  This may sound a bit abstruse and remote, so let me suggest an analogy. Imagine the challenge of writing out a clear explanation of the meaning of the contemporary term “soap digest rack” in the ancient Indian language of Sanskrit. The key constraint is that you are restricted to using pure Sanskrit as it was in its heyday, and are not allowed to introduce even one single new word into the language.

  In order to get across the meaning of “soap digest rack” in detail, you would have to explain, for starters, the notions of electricity and electromagnetic waves, of TV cameras and transmitters and TV sets, of TV shows and advertising, the notion of washing machines and rivalries between detergent companies, the idea of daily episodes of predictable hackneyed melodramas broadcast into the homes of millions of people, the image of viewers addicted to endlessly circling plots, the concept of a grocery store, of a checkout stand, of magazines, of display racks, and on and on… Each of the words “soap”, “digest”, and “rack” would wind up being expanded into a chain of ancient Sanskrit words thousands of times longer than itself. Your final text would fill up hundreds of pages in order to get across the meaning of this three-word phrase for a modern banality.

  Likewise, Gödel’s string KG, which we conventionally express in supercondensed form through phrases such as “I am not provable in PM”, would, if written out in pure PM notation, be monstrously long — and yet despite its formidable size, we understand precisely what it says. How is that possible? It is a result of its condensability. KG is not a random sequence of PM symbols, but a formula possessing a great deal of structure. Just as the billions of cells comprising a heart are so extremely organized that they can be summarized in the single word “pump”, so the myriad symbols in KG can be summarized in a few well-chosen English words.

  To return to the Sanskrit challenge, imagine that I changed the rules, allowing you to define new Sanskrit words and to employ them in the definitions of yet further new Sanskrit words. Thus “electricity” could be defined and used in the description of TV cameras and televisions and washing machines, and “TV program” could be used in the definition of “soap opera”, and so forth. If abbreviations could thus be piled on abbreviations in an unlimited fashion, then it is likely that instead of producing a book-length Sanskrit explanation of “soap digest rack”, you would need only a few pages, perhaps even less. Of course, in all this, you would have radically changed the Sanskrit language, carrying it forwards in time a few thousand years, but that is how languages always progress. And that is also the way the human mind works — by the compounding of old ideas into new structures that become new ideas that can themselves be used in compounds, and round and round endlessly, growing ever more remote from the basic earthbound imagery that is each language’s soil.

  Winding Up the Debriefing

  In my allegory, both the Klüdgerot and the Alfbert supposedly have the ability to read pure PM strings — strings that contain no abbreviations whatsoever. Since at one level (the level perceived by the Klüdgerot) these strings talk about themselves, they are like Gödel’s KG, and this means that such strings are, for want of a better term, infinitely huge (for all practical purposes, anyway). This means that any attempt to read them as statements about numbers will never yield anything comprehensible at all, and so the Alfbert’s ability, as described, is a total impossibility. But so is the Klüdgerot’s, since they too are overwhelmed by an endless sea of symbols. The only hope for either the Alfbert or the Klüdgerot is to notice that certain patterns are used over and over again in the sea of symbols, and to give these patterns names, thus compressing the string into something more manageable, and then carrying this process of patternfinding and compression out at the new, shorter level, and each time compressing further and further and further until finally the whole string collapses down into just one simple idea: “I am not edible” (or, translating out of the allegory, “I am not provable”).

  Bertrand Russell never imagined this kind of a level-shift when he thought about the strings of PM. He was trapped by the understandable preconception that statements about whole numbers, no matter how long or complicated they might get, would always retain the familiar flavor of standard number-theoretical statements such as “There are infinitely many primes” or “There are only three pure powers in the Fibonacci sequence.” It never occurred to him that some statements could have such intricate hierarchical structures that the number-theoretical ideas they would express would no longer feel like ideas about numbers. As I observed in Chapter 11, a dog does not imagine or understand that certain large arrays of colored dots can be so structured that they are no longer just huge sets of colored dots but become pictures of people, houses, dogs, and many other things. The higher level takes perceptual precedence over the lower
level, and in the process becomes the “more real” of the two. The lower level gets forgotten, lost in the shuffle.

  Such an upwards level-shift is a profound perceptual change, and when it takes place in an unfamiliar, abstract setting, such as the world of strings of Principia Mathematica, it can sound very improbable, even though when it takes place in a familiar setting (such as a TV screen), it is trivially obvious.

  My allegory was written in order to illustrate a downwards level-shift that is seen as very improbable. The Klüdgerot see only high-level meanings like “I am edible” in certain enormous PM strings, and they supposedly cannot imagine any lower-level meaning also residing in those strings. To us who know the original intent of the strings of symbols in Principia Mathematica, this sounds like an inexplicably rigid prejudice, yet when it comes to understanding our own nature, the tables are quite turned, for a very similar rigid prejudice in favor of high-level (and only high-level) perception turns out to pervade and even to define “the human condition”.

  Trapped at the High Level

  For us conscious, self-aware, “I”-driven humans, it is almost impossible to imagine moving down, down, down to the neuronal level of our brains, and slowing down, down, down, so that we can see (or at least can imagine) each and every chemical squirting in each and every synaptic cleft — a gigantic shift in perspective that would seem to instantly drain brain activity of all symbolic quality. No meanings would remain down there, no sticky semantic juice — just astronomical numbers of meaningless, inanimate molecules, squirting meaninglessly away, all the livelong, lifeless day.